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Φ[χ] = Σj exp[α∫dt φj(χ(t), t)],
[dar(f; a, a + 1/α)]pre Φ[χ]
= ∫dx f(x)* Φ[ξ(□, x)]
= ∫dx f(x)* Σj exp[α∫dt φj(ξ(t, x), t)]
= Σj ∫dx f(x)* exp[α∫aa+1/α dt φj(x, t)] exp[α∫t'＜a or t'≧a+1/α dt' φj(χ(t'), t')]
dar(f; a, a + 1/α) Φ[χ(a,a+1/α)]
= [dar(f; a, a + 1/α)]pre Φ[χ]
= Σj ∫dx f(x)* exp[α∫aa+1/α dt φj(x, t)] exp[α∫t'＜a or t'≧a+1/α dt' φj(χ(t'), t')]
= Σj ∫dx f(x)* exp[α∫aa+1/α dt φj(x, t)] exp[α∫-∞a dt' φj(χ(a,a+1/α)(t'), t') + α∫a∞ dt' φj(χ(a,a+1/α)(t'), t'+1/α)]
≒ Σj ∫dx f(x)* exp φj(x, a)
exp[ Σk＜0 φj(χ(a,a+1/α)(a + k/α), a + k/α) + Σk≧0 φj(χ(a,a+1/α)(a + k/α), a + (k + 1)/α)] ∵ α ≫ 1
dar(f; a, a + 1/α) Φ[χ]
≒ Σj ∫dx f(x)* exp φj(x, a) exp[ Σk＜0 φj(χ(a + k/α), a + k/α) + Σk≧0 φj(χ(a + k/α), a + (k + 1)/α)]
Let exp φ'j be a solution of the old Schrödinger equation for each j.
Let {exp φ'1(□, t), exp φ'2(□, t), ・・・} be an orthonormal basis of the state space of the old quantum mechanics for each t.
Let exp φj be a wave function given from exp φ'j by the twist remove normalization for each j.
∫Dχ dar(g; b, b + 1/α) Φ[χ]・{dar(f; a, a + 1/α) Φ[χ]}*
≒ Σj,j' [Πk∫dχ(a+k/α)]
∫dx' g(x')* exp φj'(x', b) exp[ Σk'＜0 φj'(χ(b + k'/α), b + k'/α) + Σk'≧0 φj'(χ(b + k'/α), b + (k' + 1)/α)]
{∫dx f(x)* exp φj(x, a) exp[ Σk＜0 φj(χ(a + k/α), a + k/α) + Σk≧0 φj(χ(a + k/α), a + (k + 1)/α)]}*
= Σj,j' ∫dx' g(x')* exp φj'(x', b) ∫dx f(x) [exp φj(x, a)]*
Πk＜0 ∫dχ(a+k/α) [exp φj(χ(a + k/α), a + k/α)]* [exp φj(χ(a + k/α), a + k/α)]
Π0≦k＜α(b-a) ∫dχ(a+k/α) [exp φj(χ(a + k/α), a + (k + 1)/α)]* [exp φj(χ(a + k/α), a + k/α)]
Πk≧α(b-a) ∫dχ(a+k/α) [exp φj(χ(a + k/α), a + k/α)]* [exp φj(χ(a + k/α), a + k/α)]
= Σj,j' ∫dx' g(x')* exp φj'(x', b) ∫dx f(x) [exp φj(x, a)]*
(δj,j')∞ Π0≦k＜α(b-a) ∫dχ(a+k/α) [exp φj(χ(a + k/α), a + (k + 1)/α)]* [exp φj(χ(a + k/α), a + k/α)]
= Σj ＜g|j,b＞＜j,a|f＞ Π0≦k＜α(b-a) ＜j,a+(k+1)/α|j,a+k/α＞
≒ Σj ＜g|j,b＞＜j,a|f＞ ∵ ※
= Σj ＜g|U(b,a)|j,a＞＜j,a|f＞
= ＜g|U(b,a)|f＞
if α(b - a) ∈ N.
---※---
Π0≦k＜α(b-a) ∫dχ(a+k/α) [exp φj(χ(a + k/α), a + (k + 1)/α)]* [exp φj(χ(a + k/α), a + k/α)]
= Π0≦k＜α(b-a) ＜j,a+(k+1)/α|j,a+k/α＞
≒ Π0≦k＜α(b-a) ＜j,a+k/α|j,a+k/α＞ ∵the twist remove normalization
= ＜j,a|j,a＞α(b-a)
= 1
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However, notice that the new grammar version of the Schrödinger equation has no solution such that Φ[χ] = Σj exp[α∫dt φj(χ(t), t)].
A few approximations caused by the finiteness of α shift the result from the one in the old quantum mechanics.
Such an approximation is not needed if α is infinity.
I don't think that it is a prediction of the new theory distinct from the old quantum mechanics.
I think that it should be understood as the fact that a quantum history and a measurement are less related to each other than in the old quantum mechanics.
A prediction of the new theory distinct from the old quantum mechanics will be caused by the entanglement of a quantum history in a time-like direction.
---
This article is a rewrite of the article 'Examine Probability Formula' in the following page.
Later Edition@Theory of Quantum History Entangled in Time-like Direction@Products of Grammatical Physics@Grammatical Physics@Forum@Vintage(2008-2014)
The content of this article was presented by me at JPS 2013 Autumn Meeting.
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